In the short term it's definitely incorrect and therefore should not be used. However, as it is quite common today, it might get accepted in the long run—who knows? But for now we should stick to depuis l'âge de X ans (or also dès l'âge de X ans in the past)…

Yes, you should use the present tense in French. When you say "I have been reading them..." it means that you still continue to read those books now, and so you need the present to talk about that in French.

The normal direct object pronoun for "them" (3rd person plural) is les, and it goes before the verb:

Je les lis...

And for the second half of your sentence, you have several options: ...depuis toute petite, depuis l'âge de 7 ans, depuis mon enfance, depuis que je suis petite, etc. (Feminine agreement, because you're a girl.)

I won't say that the passé composé is wrong, but it is less frequent... because being little wasn't a one-time event.